学术报告

题 目：Invariant Lagrangian manifolds of symplectic twist maps in all dimensions 

报告人：Marie-Claude Arnaud (University of Avignon, France)

Lecture 1: symplectic twist maps in all dimensions and their invariant Lagrangian graphs

摘要：We will introduce the symplectic twist maps on the n-dimensional annulus and a variational principle to find their orbits. 
Noting that the so-called KAM tori (we won’t prove KAM theorems, just give a statement) are Lagrangian, we will deal with a conjecture due to M. Herman.
Noting that every orbit contained in an invariant Lagrangien graph is minimizing, we will introduce the minimizing locus N. 

时间: 1月12日 周一下午15:00-17:00 

地点: 蒙民伟楼1105

Lecture 2: Green bundles and invariant minimizing submanifolds with prescribed dynamics

摘要：We will define the Green bundles along the minimizing orbits of a symplectic twist maps and prove their main properties. Using them, we will then proved that certain Lipschitz or continuously differnetiable invariant submanifolds contained in N with a prescribed dynamics are in fact Lagrangian graphs.

时间: 1月16日 周五上午9:00-11:00 

地点: 蒙民伟楼1105

Lecture 3: Birkhoff multidimensional, problems of regularity and destruction of invariant Lagrangian manifolds. 

摘要：Without proving it, we will recall the  multidimensional Birkhoff theorem due to M. Herman for twist maps and discuss the possibility to extend it. 

We will prove that with some dynamical asumptions, any invariant Lipschitz Lagrangian graph by a symplectic twist map is continuously differentiable, and some statement specific to dimension 2. 

We will then give examples of symplectic twist maps having no invariant Lagrangian graphs and discuss the possibility to destroy all the invariant tori by perturbation.

时间: 1月19日 周一上午9:00-11:00 

地点: 蒙民伟楼1105

Bibliography:

M.-C. Arnaud:

- When are the invariant submanifolds of symplectic  dynamics Lagrangian?  DCDS-A (Vol. 34, No. 5)  2014;

-Lower and upper bounds  for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of the Oseledet's splitting, Ergodic Theory and Dynamical Systems, volume 33, issue 03, (2013),  pp. 693-712 

C. Golé:

- Symplectic twist maps, Advances series in non-linear dynamics 18, World scientific

M. Herman:

-Inégalités a priori pour des tores lagrangiens invariants par des differomorphismes symplectiques, Publications mathématiques de l’IHES tome 70 (1989), p47-101

-Non-existence of Lagrangien graphs, preprint on the webpage of J.-C. Yoccoz.